When Trade Leads to Inefficient Public Good Provision: a Tax competition model

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1 When Trade Leads to Inefficient Public Good Provision: a Tax competition model Emmanuelle Taugourdeau, Abderrahmane Ziad To cite this version: Emmanuelle Taugourdeau, Abderrahmane Ziad. When Trade Leads to Inefficient Public Good Provision: a Tax competition model. Documents de travail du Centre d Economie de la Sorbonne ISSN : X <hal > HAL Id: hal Submitted on 3 Jun 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Documents de Travail du Centre d Economie de la Sorbonne When Trade Leads to Inefficient Public Good Provision: a Tax competition model Emmanuelle TAUGOURDEAU, Abderrahmane ZIAD Maison des Sciences Économiques, boulevard de L'Hôpital, Paris Cedex 13 ISSN : X

3 When Trade Leads to Inefficient Public Good Provision: a Tax competition model Emmanuelle Taugourdeau Abderrahmane Ziad December 2014 Abstract This paper analyses the tax competition mechanisms in a context of commodity trade. We show that the trade market equilibrium may restore the efficiency of the public good provision when agents from different countries have symmetric preferences. Asymmetry in preferences implies over or underprovision in public goods depending on the degree of asymmetry between countries. In both cases, the price adjustment leaves the capital stock unchanged so that the stock of capital is not affected by the taxes. Finally, we show that the centralized choice does not systematically restore the efficiency of the public good provision. Keywords: Tax competition, Nash equilibrium, Interregional Trade. JEL classification: H21, H41, E62, F12. CR, CNRS, Paris School of Economics, CES Université Paris 1, ENS Cachan, 61 av du president Wislon, Cachan Cedex. taugour@univ-paris1.fr CREM, University of Caen, France. abderrahmane.ziad@unicaen.fr. 1

4 1 Introduction Our paper revisits the tax competition literature by introducing commodity trade in a standard tax competition model. While in an integrated world, both international trade and capital mobility are important issues, only little attention has been devoted to the analysis of tax competition in a context of trade. This is the purpose of this paper which aims to analyse the consequences of trade balance on tax competition mechanisms when countries are either symmetric or asymmetric. The tax competition literature highlights the impact of capital mobility and the ajustments on the capital market that imply low taxes and the underprovision of the public goods cf Zodow & Mieszkowki 1986), Wildasin 1989), Wilson 1986)). A huge literature based on these seminal articles has extended these results in a context of labor mobility Bucovetsky & Wilson 1991), Wilson 1995)) or asymmetric countries Bucovestky 1991)). The number of competing regions on the equilibrium tax has been also analysed in Hoyt 1991) 1. Few papers have dealt with the introduction of trade in a tax competition model. Most of these papers have limited their analysis to a lump sum tax so that there is no distortionary effect of taxes on consumption cf Turnovsky 1988), Chari and Kehoe 1990)). Other papers considered a production and/or consumption tax that avoids any capital tax competition effects Devereux 1991), Devereux and Mansoorian 1992)). Another strand of the literature has dealt with the effect of tax exporting in models of trade on the public good provision. While it is widely held that tax exporting, by shifting the burden of taxes on the non residents, stimulates the provision of local public goods see for instance Oates 1972)), Wildasin 1987a) and 1987b) mitigates these results by introductiong a labor tax that leads the exported tax to raise the same incremental revenue as the non exported tax. Wildasin 1993) goes further by introducing a capital tax rate in the model to analyze the consequences of fiscal competition with interindustry trade. However the analysis is limited to the case of two regions considered as rather small compared to the world economy so that the return of capital is taken as fixed and does not serve as a channel of policy transmission. Exceptions are Wilson 1987) and Becker & Runkel 2012). In both models, the introduction of trade in a capital tax competition model crucially modifies the results of the standard tax competition literature. In line with these models, we show that trade 1 See also Wildasin & Wilson 2004) for a survey on capital tax competition literature. 2

5 may reinforce an inefficient distribution of public goods in a context of trade equilibrium. In his paper, Wilson shows that trade creates, in addition to an inefficient distribution of public goods across regions, an inefficient pattern of trade. He develops a model with a large number of regions that may produce two types of goods. The after tax return is fixed since regions are assumed to be sufficiently small so that they have a negligible impact on the capital return. We definitely depart from Wilson by considering a two country model with endogeneous capital allocation and by analyzing the impact of the trade balance equilibrium on the tax competition game. In doing so, the constant level of capital in each region results from the market equilibrium and each price is determined by the trade balance and depends on the tax rate levels. In their paper, Becker and Runkel consider the impact of transport cost in a model of trade with tax competition. Since the traded goods are perfect substitutes, there is no trade between symmetric regions at the equilibrium and even a small transport cost restores the efficiency of the public good provision. This result is due to the fact that transport cost in the product sector makes the capital sticky. In our model, we obtain a similar result in a quite different framework. We show that even without transport cost, efficiency of public good provision may be restored when trade balance is required under a case of symmetric countries. Our paper also highlights the impact of the preferences of the traded goods and the public goods on the level of public good provision. A strong taste for the public good implies an underprovision of the public good but a high asymmetry in preferences among countries for the traded good involves an overprovision of the public good through an upward distortion of the taxes due to the prices adjustment. Finally, the centralized equilibrium gives rise to additional results: while the decentralized and centralized choices perfectly match for symmetric countries, the constraint on the trade balance avoids the centralized choice to restore the efficiency of the public good provision when countries are asymmetric. These results are obtained in a two-country model where the price adjustment allowing for a trade equilibrium becomes the key element of our tax competition framework, especially when countries are asymmetric. To our knowledge, very few papers deal with fiscal coordination in case of asymmetric countries. Among them, Cardarelli and al. 2002) analyze the sustainability of a tax harmonization in repeated games. They show that a small country may benefit from deviating from the harmonized equilibrium if asymmetry between countries is large. Our analysis of the centralized equilibrium differs also from the paper by Peralta and van 3

6 Ypersele 2006) by two different aspects. First, Peralta and van Ypersele do not consider trade and second, they consider peculiar types of fiscal coordination a minimum capital tax level and a tax range). The purpose of their analysis is to determine the acceptability of these reforms. This paper is organized as follows. The second section outlines the tax competition model with trade and derives the results for symmetric countries. In section 3 we derive the Nash equilibria according to the degree of asymmetry among countries. In section 4 we characterize the centralized equilibrium and compare the results with the decentralized equilibrium. The final section summarizes our conclusions. 2 The model Consider an economy composed of two countries A and B. Each country is specialized in the production of a distinct good: jurisdiction A produces good a whereas jurisdiction B produces good b. For analytical simplicity, we assume that both jurisdictions are identical in size, and there is a single consumer in each jurisdiction who wants to consume both goods. In order to maximize each representative consumer s welfare, both jurisdictions are incited to trade with each other. In each jurisdiction, each firm uses capital to produce its output, this capital being perfectly mobile between the two jurisdictions, and some locationally fixed factor, such as land which is held entirely by the representative consumer in each jurisdiction. Each firm provides its local private good by using the same production technology with decreasing returns to scale, that is to say, an increasing, twice continuously-differentiable and strictly concave function denoted by f k i ) which depends exclusively on capital demand k i since fixed factors as explicit argument are suppressed from the production function. Capital being perfectly mobile between countries, the net of tax returns of capital equal between countries: ρ = ρ i = p i f k i ) t i = r i t i 1) where p i stands for the price of good i and t i for the tax on capital in region i. The demand for capital in country i can be rewritten as k i = k i t i +ρ. Each government provides a public good denoted by g i, i = A, B which is financed p i ] 4

7 by tax on capital. The government i s budget constraint writes: p i g i = t i k i 2) Let c A a and c A b be the quantities of good a and good b consumed by country A s representative consumer, and let c B a and c B b be the quantities consumed by country B s representative consumer. Goods a and b market equilibria write: f k A) = c A a + c B a + g A 3) and f k B) = c B b + c A b + g B 4) We assume that consumers of countries A and B have a Cobb-Douglas utility function U A and U B of the form 2 : ) c U i i ) g i 1 = with i = A, B 1 where c i is the private consumption in country i, i.e. a bundle of goods produced in each country such that: c A = ) c A α ) a c A 1 α b ; c B = ) c B β ) b c B 1 β a with α β which means that country A s household value equally or more the domestic good relative to the imported good than the country B s household. The marginal rate of substitution between the national private good and the public good in country i, hereafter denoted by MRS i is given by MRS A = U A / g A U A / c A a MRS B = U B / g B U B / c B b = 1 1 c A a α g A = 1 β 1 c B b g B 2 The general case without specifying the utility functions may be studied but with complex and demanding conditions on the primitives of the model. 5

8 so that MRS i g i < 0, MRSi c i i Country A consumer s budget constraint writes: > 0 and MRS i g i =0 > 0 for all ci i > 0 3 p a c A a + p b c A b = p a fk A ) ρ A + t A) k A + ρθ A 2k 5) and for the representative consumer in country B it becomes: p a c B a + p b c B b = p b fk B ) ρ B + t B) k B + ρθ B 2k 6) 2k being the total amount of capital in the economy. The parameter θ i stands for the proportion of the capital owned by the agent of country i and θ A + θ B = 1. Following Zodrow and Mieszkowski 1986) we consider θ A = θ B = 1 2 so that θ A2k = θ B 2k = k. Finally, each country being specialized in the production of a specific good that is consumed in both countries, the trade balance equilibrium between both countries requires: p b c A b = p a c B a 7) Inserting 2), 3) and 7) in 5) for country A, and symmetrically for country B, gives the following relation: 4 k i = k i 8) With a trade balance equilibrium, the level of the capital demand remains unchanged when the production function faces decreasing returns to scale. A change in the capital tax rates impacts both the relative price ) p a p b and the net return of capital ρ) so that the level of capital demand is not affected by capital tax changes since the effects of the relative price and the net return of capital offset. This result is in line with Becker and 3 These conditions are all assumed in Bucovetsky 1991). 4 If instead to have two different markets, which is a consequence of the fact that each country is specialized in the production of a distinct good, we have a world or a common product market, that is goods a and b are essentially the same, then market equilibria write: f k A) + f k B) = c A a + c B a + g A + c B b + c A b + g B. In this case the capital demand may be arbitrary depending on the taxes). 6

9 Runkel 2012) 5. This is a key difference with the standard tax competition models in which the capital tax base is affected by a tax rate modification. The arbitrage condition allowing for the capital market equilibrium yields: p a p b = ta t B 9) f so that the difference in prices directly depends on the difference in taxes, the marginal production being fixed due to the adjustments of the prices. The country i s representative consumer chooses his level of consumption of both goods so as to maximize his welfare function subject to his budget constraint. The maximization program in both jurisdictions gives the following relationships: α 1 α c A b c A a = p a = 1 β c B b p b β c B a and consumptions in goods a and b are given by: 10) c A b = 1 α) p b pa f t A k ] ; c A a = α p a pa f t A k ] c B a = 1 β) p a pb f t B k ] ; c B b = β p b pb f t B k ] Contrary to Wilson 1986) and Becker & Runkel 2012), relative prices are not sufficient to specify the equilibrium of the economy so that one of the price cannot be set as a numeraire. Indeed, our equilibrium requires not only the equilibrium on the capital and product markets but also on the external market through the balance of trade. Both the difference in prices and the relative prices matter in the analysis. Let us briefly develop the symmetric case, which is commonly studied in most of the tax competition models. Proposition 1. With symmetric countries, i) when C 0 holds, the symmetric Nash equilibrium tax rate is given by t A = t B = t = 1 ) f ε k and tax competition with trade induces an optimal provision of public goods in both countries. 5 In Wilson 1987), the fixed stock of capital is given by the assumptions of the model and does not result from the market equilibrium. 7

10 ii) when C 0 does not hold, the symmetric Nash equilibrium tax rate is given by t A = t B = t = f and tax competition with trade induces an under provision of public goods in both countries. with C 0 : ε k 1 and ε k = f k f capital. Proof. See Appendix 1 stands for the production function elasticity of When the production elasticity dominates the preference for the public good, the condition ensuring the balance of trade equilibrium restores the efficiency of the capital tax rates at the symmetric equilibrium. Both countries being perfectly symmetric, the balance of trade equilibrium requires t A = t B so that p a = p b and pa p b = 1. The provision of the public good is optimal MRS i = 1) because the strategic effects implied by the standard tax competition mechanism is canceled by the prices adjustment. The symmetry in preferences for the national good does not distort the external market in favor of one of the countries 6. This result is consistent with Becker and Runkel 2012) while the mechanisms allowing for the optimality of the public good provision is different. In the Becker and Rundel s paper, the price adjustment works through the existence of transport costs. Proposition 1 also highlights that the optimality of the public good provision is no longer valid when the preference for the public good dominates the production elasticity. The high taste for the public good relative to the production elasticity implies an underprovision of the public good at the equilibrium since more public goods would be desired by the agents but the arbitrage condition limits its maximum level. Note that the level of the production elasticity ε k is fixed since the level of capital remains unchanged. In addition, 1 > ε k > 0 due to decreasing returns to scale. 3 Asymmetric Countries In this section, we state α > β so that countries differ in their preferences for the national good relative to the imported good. In other words, country A s agent values more the 6 We follow the standard literature by defining the inefficient provision of public goods an allocation characterized by the inequality between the marginal rate of substitution M RS) and the marginal rate of transformation MRT ) see Atkinson and Stern 1974)). Here MRT = 1. 8

11 national good than the country B s agent. This creates an asymmetry in the trade market that may induce a kind of leadership in the fiscal decision in favor of country A. Combining Equations 5), 6), 7) and 10) we obtain: p b c A b = 1 α) p a f t A k ] = 1 β) p b f t B k ] = p a c B a 11) Condition 11) together with condition 1) given the level of capital 8) allow us to characterize the prices p a and p b as functions of the taxes: which implies p a = 1 β) t A t B) 1 ε k ) f α β) p b = 1 α) t A t B) 1 ε k ) f α β) + ε kt A f 12) ε kt B f 13) dp a dt A = dp b dt A + 1 f and dp b dt B = dp a dt B + 1 f 14) Expressions 12) and 13) show that a large asymmetry between countries impact the equilibrium prices. Moreover, for given tax rates, a large asymmetry impacts more p a than p b since it stimulates the demand for good a at the expense of good b. Since production of both goods is fixed to maintain the capital market equilibrium, prices adjust so that the trade balance stays in equilibrium. From the expressions above, we directly derive the following lemma: Lemma 1. Positive prices requires t A t B Proof. p a > 0 t A t B) > p b > 0 t A t B) > ε k 1 ε k ) ε k 1 ε k ) α β) 1 β) ta α β) 1 α) tb 0 for any t B 0 In country A, households have stronger preferences for the national good which implies a trade advantage for country A and the possibility to set a higher tax rate without supporting capital outflows or trade imbalance. As a result, the difference in prices is always positive Equation 9)). An increase in the country A s capital tax implies an increase in the difference in prices whereas an increase in the country B s capital tax implies a decrease in the difference in prices. The difference in prices adjusts so as to maintain an equal stock of capital in each country. Note that the difference 9

12 in prices does not depend on the parameters that characterize the asymmetry between countries α and β). Let us derive the impact of the capital taxes on the prices. With α > β, we obtain: d dp a dt = f A Ω α β) + 1 α) 1 ε k)) > 0; dp a 1 β) = dtb Ω f ε k 1) < 0 dp b 1 α) = dta Ω f 1 ε k) > 0; 15) dp b dt = f > B Ω α β) 1 β) 1 ε k)) = 0 16) < ) ) p a p p b = tb k 1 d a p b > 0; = ta k 1 < 0 dt A ff p 2 b dt B ff p 2 b with Ω = ff α β) > 0 The increase in tax in one country has clear-cut effects on the price of the foreign good. A rise in the tax rate of country A undoubtedly increases the price of good b by decreasing the demand from country A in good b and increasing the difference in prices to maintain the capital market equilibrium. In order to maintain the balance trade in equilibrium, p b increases. The impact of t B on p a is negative because t B tends to decrease the difference in prices. The impact of t A on p a implies an additional argument based on the asymmetry between countries: the negative direct impact on c A b is relatively smaller compared to the positive impact on c A b derived from the difference in prices. In order to adjust the balance of trade, p a has to increase. Finally, contrary to the difference in prices, the relative price reaction to a change in taxes depends on the asymmetry between countries through the level of p b. The bigger the asymmetry, the higher the relative price reaction to a change in taxes. The impact of t B on p b is not clear-cut. Let us derive the following lemma: Lemma 2. When the asymmetry between countries is strong enough, dp b dt B > 0. Proof. Directly from Equation 16) we deduce: dp b dt B > 0 α β) > 1 β) 1 ε f) 10

13 Due to the asymmetry, the negative impact on c B a following a rise in t B can be either higher or smaller than the negative impact on c A b resulting from the decrease in the difference in prices. This depends on the size of the asymmetry relative to the production elasticity. When the asymmetry between countries is high, the impact on c A b is rather limited so that the impact on c B a is relatively high and p b has to increase to reestablish the balance of trade equilibrium. The opposite result applies when the asymmetry between countries is rather small. According to our model, capital supply k is fixed, whenever the solution of 1) and 8) requires that capital earns a non-negative net return, ρ 0. When ρ < 0, we assume that capital owners do not supply any capital: Claim 1. When ρ < 0, the net return of capital is negative so that the capitalists keep their capital outside the economy. Utilities of countries A and B vanish. Before determining the optimal level of taxes we have to specify the set of strategies. According to the previous claim, the following lemma defines the sets of strategies: Lemma 3. The profile of strategies t = t A, t B) ] is defined on t B 1 β) ], T 0, 1 α) 1 α) 1 β) ta Proof. See Appendix 2. The sets of strategies are deduced from the different constraints of the model i.e. positive consumptions and non negative returns to capital. The asymmetry in preferences for the traded good allows country A to set a higher tax. With no particular restrictions, T. If restrictions on taxes or price levels are given, T can be bounded. Even if T appears to be a particular unrealistic and peculiar case, let us insist on the fact that only the difference in taxes and therefore the difference in prices and relative prices matter. Claim 2. When ρ = 0 the amount of capital in each country is equal to k. Proof. When t A > 1 β) 1 α) tb that implies ρ > 0, we have k i = k. Then lim k i = k when ] + t A 1 α) 1 β) tb so that ki = k for ρ = 0. The analysis of the marginal rate of substitution between the public and the private goods is particularly important to analyze the distortive effects of preference asymmetry on the public good provision. 11

14 Lemma 4. Asymmetric countries with α > β imply MRS B MRS A Proof. with MRSA t A > 0, MRSB t A MRS A = U A / g A U A / c A a > 0, MRSA = 1 pa f t A k ] t A k < 0 and MRSB < 0 = 1 1 β) t A t B) f f k ) t A kf α β) and MRS B = U B / g B U B / c B b = 1 pb f t B k ] t B k = 1 1 α) t A t B) f f k ) t B kf α β) 1 1 β) t A t B) f f k ) t A kf α β) MRS A > MRS B > 1 t B A 1 α) > t 1 β) 1 α) t A t B) f f k ) t B kf α β) which is a condition that cannot be satisfied according to the constraint of the model see proof of lemma 3). The signs of MRSA Finally, MRSA t A Before determining the properties of the Nash equilibrium, let us define the equilibrium: MRS, B t A t A = 1 1 β)f f k) kf α β) and MRSB t B t A ) 2 > 0 are obvious from the expressions above. Definition 1. A profile t = ] ]) t A, t B ) is a Nash equilibrium of the game Γ 2, t B 1 β), T 0, 1 α) 1 α) 1 β) ta, if none of the unilateral deviation is profitable i = A, B. Let us assume that a Nash equilibrium exists and let us denote this equilibrium by t A, t B )7. The following propositions will determine the nature of the equilibrium. Proposition 2. An equilibrium with interior solutions for both countries does not exist. 7 The question of the existence of a Nash equilibrium is very difficult and complex. In this paper we specifically focus on the impact of trade on the tax competition results. 12

15 Proof. See Appendix 3 An interior equilibrium for country A requires a marginal rate of substitution higher than 1 because an increase in the country A s capital tax implies a decrease in the public good g A p due to a high elasticity of p a t A a t A p a ). The opposite mechanism works for country B: a tax rate increase in country B implies a rise in the public good g B due to a negative or low elasticity of price p b. Hence, an interior solution in country B requires a marginal rate of substitution lower than 1. According to Lemma 4, both conditions cannot be fulfilled at the same time. As a result, one of the tax rate will be constrained by the boundary of its set of strategies. The asymmetry between countries works as an additional constraint that avoids one of the country to reach an optimal interior solution. Proposition 3. Assume that condition C 1 holds large asymmetry between countries). The asymmetric Nash equilibrium is characterized by t A = T and t B solution of V B = 0 with t A = T. where C 1 : α β) > 1 1) 1 β) 1 ) ε Proof. see Appendix 4 Because of the high price elasticity of good a, the provision of the public good in country A decreases with an increase in t A. 8 This works for the good a consumption on the good market. When the asymmetry between countries is large, the marginal rate of substitution MRS A) is lower than one and the increase in the private goods, thanks to the price adjustment, always dominates the decrease in the public good in the utility function. Then, the welfare of country A s agent increases whatever the level of t A. The country A tax rate is set to its maximum. In country B, the government limits the level of t B relative to t A since a too high level of t B would imply a higher level of public good at the expense of the private one. Corollary 1. Under condition C 1 large asymmetry between countries) tax competition with trade induces an over provision of public good in both countries. Proof. According to Lemma 2, we know that MRS B > MRS A. In Appendix 3, it is shown that an interior solution for country B implies 1 > MRS B. Both relations induce 1 > MRS B > MRS A. ) 8 dg A dt A = k p a 1 pa t A t A p a with pa t A t A p a high. 13

16 Asymmetry between countries introduces distortive mechanisms in the provision of public goods but crucially different from the ones observed in the standard tax competition models. Here, distortions are not driven by the capital market but by the external market tensions. With no trade, tax competition implies a low level of tax rates and an underprovision of public goods because of the adjustment on the capital market in order to verify the arbitrage condition. In our model, since country A s tax rate is fixed to its maximum, taxes are distorted upwards which implies an overprovision of public goods in both countries. As discussed above, a large asymmetry between countries requires strong adjustments of prices that are harmful for the economic efficiency. Proposition 4. Assume that conditions C 2 and C 3 hold small asymmetry ) between countries), the asymmetric Nash equilibrium is characterized by t A, 1 α 1 β ta with t A 0. where C 2 : ln α β) < 1 β) 1 α) ln 1 ε 1 ) f ) α ln 1 β) 1 α)ε f ]+ln 1 β) 1 α) ε f ) ) 1 and C 3 : α β) < ε k 1 1 ) 1 Proof. See Appendix 5 Both conditions C 2 and C 3 specify that the asymmetry between countries is small. A small asymmetry limits the extent of the relative price adjustment to a change in tax rates. This vanishes the unlimited positive effect of the country A tax rate on the agent s utility since a too high level of t A would imply a negative effect on welfare because a small asymmetry can no longer compensate the negative effect of the public good. Conversely, country B benefits from this small asymmetry, and now, is able to fix a tax rate on the upper bound of its set of strategies without suffering from strong price adjustments. Note that this equilibrium leads to a zero return of capital but a constant level of capital, as specified in Claim 2. Corollary 2. Under conditions C 2 and C 3 small asymmetry between countries) tax competition with trade induces an underprovision of the public good in both countries. ) Proof. When t B = 1 α 1 β ta, MRS A = MRS B 1 1 = ε f 1. Condition C 3 immediatly implies MRS B = MRS A > 1 A rather small asymmetry between countries leads to an underprovision of public goods in both countries. Either in a symmetric or an asymmetric case, when the taste 14

17 for the public good is high relative to the private good, it leads to an underprovision of the public good since the maximum tax rate is not high enough to ensure an optimal provision of the public good. 4 Centralized equilibrium In this section, we compare the decentralized with the centralized equilibrium. The centralized equilibrium aims to feature the results that would arise in the case of a centralized European government that would fix the level of the capital tax on behalf of each country. Before determining the properties of such an equilibrium, let us first discuss the symmetric case and the impossibility of tax harmonization when countries are asymmetric. First, for symmetric countries prices are equal and so the taxes) so that there is no difference between the centralized and the decentralized equilibrium. Second, for asymmetric countries tax harmonization would imply that the centralized government chooses the level of the uniform tax t A = t B = t ) that maximizes the sum of the welfare. The arbitrage condition 9) implies that the prices should be equal if a uniform tax is implemented. As a result, the trade balance cannot be in equilibrium when the preferences for the national good are asymmetric among countries Equation 11)). From now, we concentrate on the case of unequal national taxes determined at the centralized level. In the centralized equilibrium, the social planner aims to maximize the welfare of both representative consumers with respect to their strategic variables t i. The program of the social planner is the following: max V C where V C = U A + U B t A,t B Comparing the centralized results to the decentralized results leads to the following proposition Proposition 5. If V C is concave, the asymmetric centralized equilibrium implies lower or equal tax rates compared to the Nash equilibrium when either C 1 or C 2 and C 3 hold large or small asymmetry between countries). Proof. See Appendix 6. 15

18 This result is definitely different from the standard tax competition results that highlight the too low level of the tax rates at the Nash equilibrium compared to the centralized choice. An analysis of the impact of the tax rates on the marginal rate of substitution implies the following lemma: Lemma 5. A decrease resp. increase) in t A and t B implies an increase resp. decrease) in the marginal rates of substitution in both countries MRS A and MRS B ) if and only if the tax response elasticity tb dt A ) is t A dt B lower resp. higher) than 1 does not modify the marginal rates of substitution in both countries if and only if the tax response elasticity is equal to 1. Proof. See Appendix 7. Following a decrease in t B, the increase or decrease in the marginal rates of substitution depends on the tax response elasticity A t B. The tax reponse elasticity )9 dt dt B t A characterizes the response of country A s tax rate to a change of country B s tax rate. A small tax response elasticity implies a weak reaction of country A s tax rate to a change in country B s tax rate. Since MRS A responds positively to a decrease in country B s tax rate, this effect dominates the effect on MRS A of the country A s tax rate response. As a result, the MRS increases in country A. applies. Proposition 6. Compared to the Nash equilibrium; For country B the same mechanism If dta t B > 1, the centralized equilibrium worsens the overprovision of public goods dt B t A in Country A and B when C 1 holds. If dta t B < 1, the centralized equilibrium worsens the underprovision of public goods dt B t A in Country A and B when C 2 and C 3 hold. If dta t B = 1, the centralized equilibrium does not modify the provision of public dt B t A goods and inefficiencies of public goods provision remain unchanged. 9 The term tax response elasticity has been introduced by Hindriks and Nishimura 2014) by contrast with the tax base elasticity. 16

19 Proof. Using Proposition 6, we know that t A N > t A C to the centralized equilibrium we have dta dt B we can deduce that and t B N > t B C so that from the Nash > 0. Given that dmrsb dt B < 0 and dmrsa dt A > 0, dmrs B 0 dta t B dt B t 1 and A dmrsa 0 dta t B 1 17) dt B ta Finally, using Corollary 1 and 2, we obtain the Proposition 6 results. While a centralized equilibrium is supposed to limit inefficiencies by taking into account the externalities of the taxes, this result is no longer systematically valid in the particular case of asymmetric countries that we developed. This is due to the constraint on the external equilibrium which avoids both a tax harmonization to exist when countries are asymmetric, and a centralized equilibrium to reach the optimum. The introduction of such a constraint enables to avoid any tax competition by maintaining the capital level unchanged in each country; however at the expense of price adjustment that implies tax levels leading to an inefficient provision of public goods even in the centralized equilibrium. The tax response elasticity together with the degree of asymmetry is particularly crucial in determining the degree of inefficiency of the public good provision. Two cases are particularly interesting to comment. When the tax response elasticity is high and the asymmetry is small, the centralized equilibrium tends to exacerbate the overprovision of the public goods. Indeed, a high tax response elasticity implies a decrease in the marginal rate of substitution following a decrease in the tax rates because the country A s tax rate response is high in absolute value) relative to the country B s tax rate response. Conversely, when the tax response elasticity is lower that 1 and the asymmetry is large, the centralized equilibrium worsens the underpovision. Finally a tax response elasticity equal to one does not modify the inefficiencies of public good provision arising from the Nash equilibrium with a centralized choice. Note that we are not able to give clear-cut results when both the response elasticity and th asymmetry are high or low. Indeed, the centralization may either diminish the inefficiency of the public good provision or imply an overprovision of public good while the Nash equilibrium involved an underprovision and vice versa). 17

20 5 Conclusion In this paper, we developed a capital tax competition model with trade. We show that the trade balance equilibrium crucially modifies the tax competition mechanisms by maintaining the level of capital unchanged between countries. This may lead to an optimal level of public good provision in a symmetric countries framework. When agents have asymmetric preferences among countries, the Nash equilibrium is conditional to the degree of asymmetry. It may imply either an overprovision or an underprovision of public goods. The link between the trade market and the capital market creates pressure on prices that are not investigated in standard tax competition models. In concordance with several papers that mitigate the benefit of fiscal cooperation among asymmetric countries, we show that a centralized choice may worsen the inefficient provision of public goods. In light with these papers, another conclusion arising from our work is that inefficiencies of public good provision may be mitigated by a centralized choice, but at the expense of trade imbalances. If we try to apply our results for European countries, our paper shows that a centralized choice will not mitigate the inefficiency of public goods provision for asymmetric countries for example Germany and Ireland) if the tax response elasticity is rather small or equal to one; a scenario which is the more consistent with the European case. The mitigation of the inefficiency may apply under trade imbalances. 6 Appendix 6.1 Appendix 1 Let us assume that α = β so that countries are perfectly symmetric. The constraint on the balance of trade together with the arbitrage condition imply p a p b = t A t B) k f = t A t B) 1 f so that for any f k f we have p a = p b, t A = t B = t and g A = g B = g. We deduce 18

21 the expressions of the private and public consumptions as g A = g B = tk p c A a = c B b = α f tk ) p c A b = c B a = 1 α) f tk ) p Normalizing the common price to the unity p = 1), we deduce that the indirect utility functions denoted by V A V A t) and V B V B t) with t 0, f ] according to the arbitrage condition. They can be rewritten as V i = U i f g c j i, cj i, g] with i = A, B, j = A, B and i j The symmetric Nash equilibrium is obtained by maximizing the indirect utility function of both countries: V i t = U i c i i dg dt + U i dg g dt = U i dg MRS i 1 ) = 0 c i i dt so that Since MRS i = 1 MRS i = U i / g i U i / c i i = 1 f g g MRS i = 1 => t = 1 ) f k Let us check that t = 1 ) f k is an interior solution. 1 ) f k f ε + 1 When ε + < 1, MRS i > 1 t 0, f ] and the Nash solution is on the boundary so that t = f. To complete the proof we have to check the concavity of the indirect utility functions: 19

22 At the equilibrium we have V )2 t 2 = U i dg MSR i c i i dt t condition ensures that there exists only one maximum. = U i c i i dg dt ) < 0. This g 6.2 Appendix 2 A non negative return of capital ρ 0) implies p a f t A and p b f t B and replacing p a and p b gives the same constraint: t A Consumptions in goods a and b are given by: 1 β) 1 α) tb c A b = 1 α) p b pa f t A k ] ; c A a = α p a pa f t A k ] c B a = 1 β) p a pb f t B k ] ; c B b = β p b pb f t B k ] According to the budget constraint of the household, positive consumptions in both countries require p a f t A k and p b f t B k Replacing p a and p b by their expressions 12) and 13) leads to the same constraint for both countries: t B t A) f f k ) 0 t B t A 18) Compiling these conditions gives: t A 1 β) 1 α) tb > t B 0 and 0 t B 1 α) 1 β) ta < t A 20

23 6.3 Appendix 3 In the case of a non-cooperative game, each government i, i = A, B aims to maximize the welfare of its representative consumer with respect to its strategic variable t i, taking the tax rate of the other government as given. V A t A V B = U A c A a = U B c B b For country A, 19) rewrites with and V A t A = U A c A a dg A dt A 1 p a c A b dg B dt B 1 p b c B a ga t A MRS A 1 ) 1 p a c A b dpb dt p )] b dp a + U A dg A 19) A p a dt A g A dt A dpa dt p )] a dp b + U B dg B 20) A p b dt A g B dt B pb t p ) b p a = β α) tb k < 0 A p a t A Ωp a g A = 1 β) tb k f k f ) t A Ω p a ) 2 < 0 An interior solution with positive consumptions requires MRS A c A b α β) = β) f f k ) > 1 pb t p )] b p a A p a t A For country B we obtain V B = U B c B b g B MRS B 1 ) 1 p b c B a pa t p )] a p b B p b with and pa t p ) a p b = β α) ta k < 0 B p b Ωp b g B = 1 α) ta k f f k ) Ω p b ) 2 > 0 21

24 An interior solution with positive consumptions requires MRS B = 1 + c B a α β) 1 α) f k f) < 1 Then an interior solution in both countries requires which is inconsistent with Lemma 4. MRS A > MRS B 6.4 Appendix 4: For country A, 19) rewrites with V A t A = U A c A a ga t A MRS A 1 ) 1 p a c A b pb t p ) b p a = α β) tb k < 0 A p a t A p a Then if MRS A < 1, we have V A t A > 0 t A for t B 0 pb t p )] b p a A p a t A MRS A = = 1 ) t B t A) 1 β) f f k ) t A k f β α) t A t B) ) 1 β) 1 1 ) t A β α) ε 1 since ta t B ) < 1, a sufficient condition that ensures MRS A < 1 t A is 1 β) 1 1) 1 ) < t A α β) ε 1. Let us denote this condition by C 1 and rewrite it as: t A 1 α) C 1 : α β > ) 1 1 ) ε 1 1 β) The best reply for country B is neither t B = 0 which implies V B = 0, nor t B = which implies 1 β) ta = t is interior. Let us rewrite V B B 1 β 1 α which contradicts C 1. Then the best response t B 22

25 V B = U ) B α) t A f k f ) c B b p b t B f β α) p a Ω 1 ) 1 α) t B t A) f f k ) t B kf β α) ) p a Ω Let denote by Z t B) t B k 1 β) t B t A) ff α β) 2] Z t B) = 1 ) 1 α) t B t A) f f k ) t B kf β α) ) p a Ω 1 β) t B t B t A) fkf α β) 2] Z t B) is a second degree polynomial of the form so that Z t B) = 0 admits only two roots. Z t B) = z 1 t B2 + z 2 t B + z 3 The existence of two maxima is impossible since a minimum would necessarily exist between the maxima and ] more than two roots would exist. The best response t B is then unique and t B 0,. 1 α) 1 β) ta 6.5 Appendix 5: Let us consider that C 1 does not hold. From Proposition 1 we know that an equilibrium with two interior solutions does not exist. Then at least one strategy of the equilibrium must be on the boundary. 1. t B = 0 implies V B = 0 so that t B = 0 is dominated by any strategies which insure V B Let analyze the case t A = T Without particular restrictions, t A is defined on values of V A for t A = t B 1 β) 1 α) and ta. t B 1 β) 1 α), ]. Let us compare the 23

26 When t A = t B 1 β) 1 α) we have p a = ta f and p b = tb f Then the utility writes g A = f k c A a = α f f k ] and c A a = 1 β) f f k ] ) V A B 1 β) α α 1 β) 1 α f f k ) ) ) f 1 k t = 1 α) 1 When t A we have lim t A Then the utility writes lim V A = t A lim t A p a p b = 1 β) f 1 α) f k 1 β) f 1 β) f k > 1 lim g A = t A ff α β) k 1 β) f 1 α) f k c A f f k a = α 1 β) f 1 β) f 1 α) f k f α 1 β) f f k 1 β)f 1 α)f k ) α 1 β) 1 α and lim c A t A a = 1 α) f f f α β) k 1 ) 1 β) f 1 α) f k And we have ) V A B 1 β) t > lim V A 1 α) t A ) α ) ) 1 α ) 1 1 β) 1 α β) < 1 1 β) 1 α) ε f 1 ε f ) 1 β) 1 α) ε f ) 1 24

27 This last condition can be rewritten as: ] 1 β) C 2 : α ln 1 α) ln 1 ε f ) 1 β) 1 α) ε f +1 ) ln α β) 1 β) 1 α) ε f < 0 ) ) Then under Condition C 2, V A t B 1 β) > lim V A and since V A t B 1 β) is of 1 α) t A 1 α) constant value whatever the value of t B, then there exists a finite T such that for any t A > T, lim V A is strictly dominated by V A t A t B 1 β) 1 α) 3. Let us assume t A = 1 β 1 α tb. At this point we know that we have V A t A t A, t B ) < 0 which implies V A t A, t B ) = U ) A 2 1 t B2 f f k ) 1 β) t A c A a p a f f p b 1 ) f f k )) + kf β α 1) ] < 0 ). 1 ) f f k )) + kf β α 1) > > ε f 1 ) 1 + α β) : C 3 According to Appendix 3, an interior solution ) for B implies MRS B < 1. For t A = 1 β 1 α tb, we obtain MRS B 1 1 = ε f 1 > 1 under C 3 which eliminates any interior solution for t B. Let us check that we also have V B t t A, t B ) > 0 B V B t A, t B ) = U B 1 α) f k f ) ) 1 β c B b 1 α 1 ) f f k ) kf ) kf β α) ] > 0 1 ) f f k ) kf 1 + β α) > > 1 + β α) ε k 1 ) which is always true under C 3. As a result, under C 3, when a Nash equilibrium exists, it is given by t A, 1 α 1 β ta ). 25

28 6.6 Appendix 6 The first derivative of the indirect utility functions V C are: V C t A = U A c A a U B c B b ga MRS A 1 ) 1 c A pb b t A p a t p )] b p a + A p a t A g B MRS B 1 ) 1 c B pa t A a p b t p )] a p b A p b t A V C = U B c B b U A c A a g B MRS B 1 ) 1 c B pa a p b t p )] a p b + B p b t B ga MRS A 1 ) 1 c A pb b p a t p )] b p a B p a If C 1 holds, V C V C evaluated at the Nash equilibrium, gives = U A g A MRS A 1 ) 1 c A Nash b p a c A a pb t p )] b p a B p a 21) with p b p b p a p a > 0; MRS A < 1; dg A dt B = ta k p a) 2 dp a dt B If a centralized equilibrium exists and V C is concave, then V S t B N > 0; so that V S t B N ) < 0. t B N ) < 0 implies that > t B C. Since the Nash equilibrium implies the highest tax rate for country A, the tax rate at the centralized equilibrium cannot be higher t A N If C 2 and C 3 hold, V C V C t A since gb t A 0. t A C ). evaluated at the Nash equilibrium, gives = U A ga MRS A 1 ) 1 c A pb Nash c A b a t A p a t p )] b p a + A p a t }{{ A } <0 U B g B MRS B 1 ) 1 c B pa c B b t A a p b t p )] a p b < 0 A p b t A = 1 α f k α β f 1 t f k f ) < 0, MRS B > 1 and A 26 ) p a pa p b t A p b = pa t A p b p b p t A b p a p a t A ) >

29 Under the assumption that V C is concave, we can] deduce that t A N ] > t A S. Finally, the collective utility V C is defined on t B 1 β), T 0, 1 α) 1 α) 1 β) ta. Under C 2 and C 3, we have t B N = 1 α 1 β t A N > 1 α 1 β t A 1 α C where 1 β t A C is the upper level that tb could take in response to t A C. This implies that t B N > t B 6.7 Appendix 7 According to Lemma 1 and its proof we can write MRS A t A = 1 1 β) f f k ) kf α β) so that the total derivative of MRS A is given by dmrs A = MRSA t A with MRSA t A > 0. For country B MRS B C. t B t A ) 2 = t B MRSA t A ) dt A + MRSA dt B = MRSA t A t B dt A t A t B dtb t A dt 1 B = 1 dmrs B = MRSB 1 α) f f k ) kf α β) dt B + MRSB t A with MRSB < 0. For tb dt A = 1, dmrs B = 0 and dmrs A = 0. t A dt B t A t B ) 2 = t B MRSB t A ) dt A = MRSB dt 1 B tb dt A t A dt B References 1] Atkinson A. & N. H. Stern, 1974): Pigou, Taxation and Public Goods, Review of Economic Studies, 41, ] Becker, J. & M. Runkel, 2012): Even small trade costs restore efficiency in tax competition, Journal of Urban Economics, 72,

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